Since a Cartesian product of zero entropy systems has zero entropy, \(S_1 \times \cdots \times S_l\)  has zero entropy.
It is known ([1]}) that natural extensions of exact transformations are K-automorphisms and a Cartesian product of K-automorphisms is K-automorphism. Thus, \(T_1 \times \cdots \times T_m\)  is a factor of a K-automorphism.
It is also known that K-automorphisms are disjoint from zero entropy systems. Also, if \(\mathsf {X}\)  and \(\mathsf {Y}\)  are disjoint and \(\mathsf {Z}\)  is a factor of \(\mathsf {X}\) , then \(\mathsf {Z}\)  and \(\mathsf {Y}\)  are disjoint (Proposition I.1 in [2]}). Thus, \(S_1 \times \cdots \times S_l\)  and \(T_1 \times \cdots \times T_m\)  are disjoint, and so Theorem REF  follows from REF .
